There seems to a good bit of disagreement among dog breeders about Hardy-Weinburg Equilibrium and
how, or whether, this mathematical model of population genetics applies to
purebred dogs. George Padgett says it's not really applicable, but it's
the best we have and we should use it anyway, even though it doesn't really
work. This is a rather iffy argument, it seems to me. If it doesn't
work, then it really doesn't, and trying to use it is likely to seriously
mislead breeders.

Jackie Isabell says it is applicable and can be used to give us
reliable information.

One can easily figure out where these authors are coming from. Here's
the thing: in order to use HWE math, the following assumptions must be
met:

1. Mating within the
population must be random.

2. Natural / artificial
selection must not be occurring.

3. Mutations must not
be occurring.

4. There must be no
immigration / emigration of individuals into or out of the population.

5. Genetic drift must
not be occurring.

Now, do populations of purebred dogs meet these criteria?

No, they obviously don't. But neither do natural populations. But
natural populations come close enough to allow the use of HWE math. Do
purebred dogs come "close enough"?

Isabell, for example, says yes -- because although mating within purebred dogs is not
random, it may be random with respect to the trait in question, and that's what
counts. Similarly, although selection is occurring, she points out that it
is usually not selection with regards to the trait in question.

Although she is right as far as that goes, Isabell is missing the fact that
HWE also implicitly assumes that each individual in the population has an equal
chance to reproduce -- that's what that random-mating requirement is really all
about -- and in addition, she does not appear to notice that popular-sire
syndrome produces very powerful genetic drift (changes in allele frequency due
to chance). In small populations, as for any of the rarer breeds, drift is
usually powerful anyway, even disregarding popular-sire syndrome.

Willis, I think, gets closer when he suggests that you can always use HW math to give
you a "snapshot" picture of the incidence of heterozygotes for particular traits
in a breed in a specific year, but unless the breed is actually at equilibrium,
the incidence of carriers will not remain constant over generations.
However, even this suggestion is put at question by
Jerold Bell.

According to Dr. Jerold Bell (the article is "Epidemiological studies of
inherited disorders" -- a copy can be found at
http://www.papillonclub.org/PapillonHealth/Article-Epidemiological-Studies.html),
this question has been answered by generational studies of genetic disorders in
domestic animals and the answer to whether HWE calculations apply to domestic
dogs is No. The actual quote is "While it is recognized that the frequency
of carriers of recessive defective genes will far exceed that of affected
individuals, there is no mathematical relationship between the two in domestic
animal breeding."

Bell is the single geneticist I respect most and though I'd be interested in
his references, I take him at his word.

HWE does not apply to dogs and cannot be used as though it did.
Padgett's estimates of carrier frequency -- anybody's estimates if they are
using HWE -- are almost certainly wrong. Isabell is wrong. Even
Willis, by this statement, seems to be wrong.

Bell recommends using "population-wide genetic testing" and pedigree analysis
to obtain accurate estimates of carrier frequencies in show dog populations.

If you're interested in the topic despite Bell's opinion, you may be asking
yourself:

But what is Hardy-Weinburg Equilibrium,
exactly?

Here's what all these authors are talking about. Notice that HWE is a
population genetics model, which may explain why genetic counselors and
so forth may not be quite up on the details of how this model is put together
and meant to be applied.

The Hardy-Weinburg principle is a mathematic model used to estimate allele and genotype frequencies in
natural populations. It states that allele and genotype frequencies remain
constant over time in populations that meet the criteria listed above. It's meant to be used to estimate genotype
frequencies for simple traits in which genes have only two possible alleles (but
can be expanded for use even if there are more than two alleles), and what dog
breeders try to do, evidently in vain if you believe Bell, is use this model to estimate the number of heterozygotes (carriers)
of a recessive trait that exist in a
population. It does not have anything to do with identifying which
specific individuals are carriers, just with estimating the overall number of
carriers for the whole population. The math involved in this particular
use of the principle is simple.

It works like this:

Let p be the frequency of the normal allele (A)

Let q be the frequency of the recessive abnormal allele (a)

Let P be the frequency of the homozygous dominant genotype (AA)

Let H be the frequency of the heterozygous (carrier) genotype (Aa)

Let Q be the frequency of the homozygous recessive genotype (aa)

Then, if HWE assumptions hold,

P = p2

H = 2pq

Q = q2

p + q = 1

Given the above, it's pretty easy to estimate everything if you can start out
knowing the frequency of affected individuals in the population.

Suppose that an extensive survey shows that 2% of all the wolves in Alaska
are pure white and that this color is believed to be a simple recessive in a
two-allele one-gene system. If 2% of wolves are
white, what proportion
of the wolves are carriers for this trait?

Q = q2 = 0.02 so
√q2 = √0.02
so q = √0.02 so q = 0.14

Now we know the frequency of the recessive
allele (a). That doesn't yet give us the frequency of carriers in the
population, but it will, because:

p + q = 1 so
p = 1 - q so p = 1 - 0.14
so p = 0.86

and

H = 2pq so
H = 2 (0.86) (0.14) so H = 0.24

and that is the carrier frequency for the
population. 24% of the wolves in this population are carriers for the
white color. Notice that a small percentage of affected animals (2%)
implies a quite large percentage of carriers (24%). Provided that the population is in Hardy-Weinburg
Equilibrium, which most natural populations are, that percentage will be the same for the next generation, and the
next -- unless natural selection selects against this color trait, at which point the
breed will no longer be at equilibrium and the frequency of the recessive allele
q will start to decline. Or selection favors the trait, in which case, of
course, the frequency of q will start to increase. Or the population of
wolves deviates substantially from one or more of the other HWE criteria.

I used wolves for this example, of course,
because they would count as a natural population and allow me to explain HWE
without regards to any questions about whether any of this applies to domestic
dogs. Let me repeat:

"While it is recognized that the frequency
of carriers of recessive defective genes will far exceed that of affected
individuals, there is no mathematical relationship between the two in domestic
animal breeding."